# L11n32

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n32 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{t(1) t(2)^5-4 t(1) t(2)^4+5 t(1) t(2)^3-t(2)^3-t(1) t(2)^2+5 t(2)^2-4 t(2)+1}{\sqrt{t(1)} t(2)^{5/2}}$ (db) Jones polynomial $-q^{5/2}+2 q^{3/2}-4 \sqrt{q}+\frac{6}{\sqrt{q}}-\frac{7}{q^{3/2}}+\frac{7}{q^{5/2}}-\frac{7}{q^{7/2}}+\frac{5}{q^{9/2}}-\frac{4}{q^{11/2}}+\frac{1}{q^{13/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $a^7 (-z)+a^5 z^5+4 a^5 z^3+4 a^5 z+2 a^5 z^{-1} -a^3 z^7-5 a^3 z^5-9 a^3 z^3-10 a^3 z-4 a^3 z^{-1} +2 a z^5+8 a z^3-z^3 a^{-1} +8 a z+3 a z^{-1} -3 z a^{-1} - a^{-1} z^{-1}$ (db) Kauffman polynomial $-a^3 z^9-a z^9-4 a^4 z^8-6 a^2 z^8-2 z^8-5 a^5 z^7-5 a^3 z^7-a z^7-z^7 a^{-1} -2 a^6 z^6+13 a^4 z^6+24 a^2 z^6+9 z^6+18 a^5 z^5+32 a^3 z^5+19 a z^5+5 z^5 a^{-1} +2 a^6 z^4-13 a^4 z^4-27 a^2 z^4-12 z^4-4 a^7 z^3-25 a^5 z^3-44 a^3 z^3-31 a z^3-8 z^3 a^{-1} -a^8 z^2-a^6 z^2+7 a^4 z^2+13 a^2 z^2+6 z^2+2 a^7 z+14 a^5 z+24 a^3 z+17 a z+5 z a^{-1} -a^6-2 a^4-3 a^2-1-2 a^5 z^{-1} -4 a^3 z^{-1} -3 a z^{-1} - a^{-1} z^{-1}$ (db)

### Khovanov Homology

The coefficients of the monomials $t^rq^j$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).
 \ r \ j \
-5-4-3-2-101234χ
6         11
4        1 -1
2       31 2
0      31  -2
-2     43   1
-4    44    0
-6   33     0
-8  24      2
-10 23       -1
-12 3        3
-141         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory. See A Sample KnotTheory Session.